You have seen that the math library contains a function that computes the
square root of numbers. In this exercise, you are to write your own algorithm
for computing square roots. One way to solve this problem is to use a guess-
and-check approach. You first guess what the square root might be, and then
see how close your guess is. You can use this information to make another
guess and continue guessing until you have found the square root (or a close
approximation to it). One particularly good way of making guesses is to use
Newton's method. Suppose \(x\) is the number we want the root of, and guess is
the current guessed answer. The guess can be improved by using computing the
next guess as:
\\[
\frac{g u e s s+\frac{x}{g u e s s}}{2}
\\]
Write a program that implements Newton's method. The program should prompt the
user for the value to find the square root of \((x)\) and the number of times to
improve the guess. Starting with a guess value of \(x / 2,\) your program should
loop the specified number of times applying Newton's method and report the
final value of guess. You should also subtract your estimate from the value of
math.sqrt (x) to show how close it is.